"Rangesize" here means "number of areas in a geographic
LAGRANGE cladogenesis model requires
that, during cladogenesis events, one daughter lineage
will ALWAYS have a geographic range of size 1. This is
argued for in Ree et al. (2008) on the grounds
that new species usually get isolated and start in a new
area. This is a reasonable proposition, but still, it
would be nice to test the assumption. In addition, it
could be that some speciation modes, especially
vicariance, obey different rules. E.g.,
(Ronquist (1996), Ronquist (1997)) allows
vicariant speciation to divide up the ancestral range in
every possible way (e.g., ABCD–>AB|CD, or AC|BD, or
A|BCD, or D|ABC, etc.), but
LAGRANGE would only
allow vicariance to split off areas of size 1:
(ABCD–>A|BCD, B|ACD, etc.) (Ronquist_Sanmartin_2011).
The maximum number of areas possible allowed for the smaller-ranged-daughter in this type of cladogenesis/speciation.
The parameter describing the probability distribution on descendant rangesizes for the smaller descendant. See above.
The output matrix consists of ancestral
rangesizes and rangesizes of the smaller descendant.
Some values are disallowed – e.g. descendant ranges
larger than the ancestor; or, in subset speciation,
descendant ranges the same size as the ancestor are
disallowed. All disallowed descendant rangesizes get
To test different models, the user has to have control of the relative probability of different descendant rangesizes. The probability of each descendant rangesize could be parameterized individually, but we have a limited amount of observational data (essentially one character), so efficient parameterizations should be sought.
One way to do this is with the Maximum Entropy (Harte (2011)) discrete probability distribution of a number of ordered states. Normally this is applied (in examples) to the problem of estimation of the relative probability of the different faces of a 6-sided die. The input "knowledge" is the true mean of the dice rolls. If the mean value is 3.5, then each face of the die will have probability 1/6. If the mean value is close to 1, then the die is severely skewed such that the probability of rolling 1 is 99 other die rolls is very small. If the mean value is close to 6, then the probability distribution is skewed towards higher numbers.
BioGeoBEARS, we use the same Maximum
Entropy function to specify the relative probability of
geographic ranges of a number of different rangesizes.
This is merely used so that a single parameter can
control the probability distribution – there is no
MaxEnt estimation going on here. The user specifies a
value for the parameter
between 0.0001 and 0.9999. This can then be applied to
all of the different ancestor-descendant range
combinations in the cladogenesis/speciation matrix.
Example values of
maxent_constraint_01 would give
the following results:
maxent_constraint_01 = 0.0001 – The smaller
descendant has rangesize 1 with 100
maxent_constraint_01 = 0.5 –
The smaller descendant can be any rangesize equal
probability. This is effectively what happens in
DIVA's version of vicariance speciation
maxent_constraint_01 = 0.9999 – The smaller
descendant will take the largest possible rangesize for a
given type of speciation, and a given ancestral
rangesize. E.g., for sympatric/range-copying speciation
(the ancestor is simply copied to both descendants, as in
a continuous-time model with no cladogenesis effect), an
ancestor of size 3 would product two descendant lineages
of size 3. Such a model is implemented in the program
BayArea (Landis et al. (2013)).
LAGRANGE, on the other hand, would only allow
range-copying for ancestral ranges of size 1.
LAGRANGE-type models, at
speciation/cladogenesis events, one descendant daughter
branch ALWAYS has size 1, whereas the other descendant
daughter branch either (a) is the same (in
sympatric/range-copying speciation), (b) inherits the
complete ancestral range (in sympatric/subset speciation)
or (c) inherits the remainder of the range (in
vicariant/range-division speciation). LAGRANGE-type
behavior (the smaller descendant has rangesize 1 with
rangesize) can be achieved by setting the
maxent_constraint_01 parameter to 0.0001.
See also: Maximum Entropy probability distribution
for discrete variable with given mean (and discrete
uniform flat prior)
Currently, the function
maxent from the
FD package is used to get the discrete
probability distribution, given the number of states and
maxent_constraint_01 parameter. This could
also be done with
get_probvals, which uses
following equations 6.3-6.4 of Harte (2011),
although this is not yet implemented.
relprob_subsets_matrix, a numeric matrix giving
the relative probability of each rangesize for the
smaller descendant of an ancestral range, conditional on
the ancestral rangesize.
Nicholas J. Matzke email@example.com
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testval=1 # Examples # Probabilities of different descendant rangesizes, for the smaller # descendant, under sympatric/subset speciation # (plus sympatric/range-copying, which is folded in): relative_probabilities_of_subsets(max_numareas=6, maxent_constraint_01=0.0001, NA_val=NA) relative_probabilities_of_subsets(max_numareas=6, maxent_constraint_01=0.5, NA_val=NA) relative_probabilities_of_subsets(max_numareas=6, maxent_constraint_01=0.9999, NA_val=NA) # Probabilities of different descendant rangesizes, for the smaller descendant, # under vicariant speciation relative_probabilities_of_vicariants(max_numareas=6, maxent_constraint_01v=0.0001, NA_val=NA) relative_probabilities_of_vicariants(max_numareas=6, maxent_constraint_01v=0.5, NA_val=NA) relative_probabilities_of_vicariants(max_numareas=6, maxent_constraint_01v=0.9999, NA_val=NA)
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